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<META name=vsisbn content="1576101746">
<META name=vstitle content="Michael Abrash's Graphics Programming Black Book, Special Edition">
<META name=vsauthor content="Michael Abrash">
<META name=vspublisher content="The Coriolis Group">
<META name=vspubdate content="07/01/97">
<META name=vscategory content="Web and Software Development: Game Development,Web and Software Development: Graphics and Multimedia Development">
<TITLE>Michael Abrash's Graphics Programming Black Book Special Edition: 3-D Shading</TITLE>
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<P><BR></P>
<H2><A NAME="Heading1"></A><FONT COLOR="#000077">Chapter 54<BR>3-D Shading
</FONT></H2>
<H3><A NAME="Heading2"></A><FONT COLOR="#000077">Putting Realistic Surfaces on Animated 3-D Objects</FONT></H3>
<P>At the end of the previous chapter, X-Sharp had just acquired basic hidden-surface capability, and performance had been vastly improved through the use of fixed-point arithmetic. In this chapter, we&#146;re going to add quite a bit more: support for 8088 and 80286 PCs, a general color model, and shading. That&#146;s an awful lot to cover in one chapter (actually, it&#146;ll spill over into the next chapter), so let&#146;s get to it!
</P>
<H3><A NAME="Heading3"></A><FONT COLOR="#000077">Support for Older Processors</FONT></H3>
<P>To date, X-Sharp has run on only the 386 and 486, because it uses 32-bit multiply and divide instructions that sub-386 processors don&#146;t support. I chose 32-bit instructions for two reasons: They&#146;re much faster for 16.16 fixed-point arithmetic than any approach that works on the 8088 and 286; and they&#146;re much easier to implement than any other approach. In short, I was after maximum performance, and I was perhaps just a little lazy.
</P>
<P>I should have known better than to try to sneak this one by you. The most common feedback I&#146;ve gotten on X-Sharp is that I should make it support the 8088 and 286. Well, I can take a hint as well as the next guy. Listing 54.1 is an improved version of FIXED.ASM, containing dual 386/8088 versions of <B>CosSin(), XformVec()</B>, and <B>ConcatXforms()</B>, as well as <B>FixedMul()</B> and <B>FixedDiv()</B>.</P>
<P>Given the new version of FIXED.ASM, with <B>USE386</B> set to 0, X-Sharp will now run on any processor. That&#146;s not to say that it will run fast on any processor, or at least not as fast as it used to. The switch to 8088 instructions makes X-Sharp&#146;s fixed-point calculations about 2.5 times slower overall. Since a PC is perhaps 40 times slower than a 486/33, we&#146;re talking about a hundred-times speed difference between the low end and mainstream. A 486/33 can animate a 72-sided ball, complete with shading (as discussed later), at 60 frames per second (fps), with plenty of cycles to spare; an 8-MHz AT can animate the same ball at about 6 fps. Clearly, the level of animation an application uses must be tailored to the available CPU horsepower.</P>
<P>The implementation of a 32-bit multiply using 8088 instructions is a simple matter of adding together four partial products. A 32-bit divide is not so simple, however. In fact, in Listing 54.1 I&#146;ve chosen not to implement a full 32&#215;32 divide, but rather only a 32&#215;16 divide. The reason is simple: performance. A 32&#215;16 divide can be implemented on an 8088 with two <B>DIV</B> instructions, but a 32&#215;32 divide takes a great deal more work, so far as I can see. (If anyone has a fast 32&#215;32 divide, or has a faster way to handle signed multiplies and divides than the approach taken by Listing 54.1, please drop me a line care of the publisher.) In X-Sharp, division is used only to divide either X or Y by Z in the process of projecting from view space to screen space, so the cost of using a 32&#215;16 divide is merely some inaccuracy in calculating screen coordinates, especially when objects get very close to the Z = 0 plane. This error is not cumulative (that is, it doesn&#146;t carry over to later frames), and in my experience doesn&#146;t cause noticeable image degradation; therefore, given the already slow performance of the 8088 and 286, I&#146;ve opted for performance over precision.</P>
<P>At any rate, please keep in mind that the non-386 version of <B>FixedDiv()</B> is <I>not</I> a general-purpose 32&#215;32 fixed-point division routine. In fact, it will generate a divide-by-zero error if passed a fixed-point divisor between -1 and 1. As I&#146;ve explained, the non-386 version of <B>Fixed-Div()</B> is designed to do just what X-Sharp needs, and no more, as quickly as possible.</P><P><BR></P>
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<font face="Verdana,sans-serif" size="1">Graphics Programming Black Book &copy; 2001 Michael Abrash</font>
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<META name=vstitle content="Michael Abrash's Graphics Programming Black Book, Special Edition">
<META name=vsauthor content="Michael Abrash">
<META name=vspublisher content="The Coriolis Group">
<META name=vspubdate content="07/01/97">
<META name=vscategory content="Web and Software Development: Game Development,Web and Software Development: Graphics and Multimedia Development">
<TITLE>Michael Abrash's Graphics Programming Black Book Special Edition: 3-D Shading</TITLE>
<!-- HEADER -->
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<!--ISBN=1576101746//-->
<!--TITLE=Michael Abrash's Graphics Programming Black Book Special Edition//-->
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<P><BR></P>
<P><B>LISTING 54.1 FIXED.ASM</B></P>
<!-- CODE //-->
<PRE>
; Fixed point routines.
; Tested with TASM
USE386 equ 1 ;1 for 386-specific opcodes, 0 for
; 8088 opcodes
MUL-ROUNDING-ON equ 1 ;1 for rounding on multiplies,
; 0 for no rounding. Not rounding is faster,
; rounding is more accurate and generally a
; good idea
DIV-ROUNDING-ON equ 0 ;1 for rounding on divides,
; 0 for no rounding. Not rounding is faster,
; rounding is more accurate, but because
; division is only performed to project to
; the screen, rounding quotients generally
; isn't necessary
ALIGNMENT equ 2
.model small
.386
.code
;=====================================================================
; Multiplies two fixed-point values together.
; C near-callable as:
; Fixedpoint FixedMul(Fixedpoint M1, Fixedpoint M2);
FMparms struc
dw 2 dup(?) ;return address &amp pushed BP
M1 dd ?
M2 dd ?
FMparms ends
align ALIGNMENT
public _FixedMul
_FixedMul proc near
push bp
mov bp,sp
if USE386
mov eax,[bp+M1]
imul dword ptr [bp+M2] ;multiply
if MUL-ROUNDING-ON
add eax,8000h ;round by adding 2^(-17)
adcedx,0 ;whole part of result is in DX
endif ;MUL-ROUNDING-ON
shreax,16 ;put the fractional part in AX
else;!USE386
;do four partial products and
; add them together, accumulating
; the result in CX:BX
push si ;preserve C register variables
push di
;figure out signs, so we can use
; unsigned multiplies
sub cx,cx ;assume both operands positive
mov ax,word ptr [bp+M1+2]
mov si,word ptr [bp+M1]
and ax,ax ;first operand negative?
jns CheckSecondOperand ;no
neg ax ;yes, so negate first operand
neg si
sbb ax,0
inc cx ;mark that first operand is negative
CheckSecondOperand:
mov bx,word ptr [bp+M2+2]
mov di,word ptr [bp+M2]
and bx,bx ;second operand negative?
jns SaveSignStatus ;no
neg bx ;yes, so negate second operand
neg di
sbb bx,0
xor cx,1 ;mark that second operand is negative
SaveSignStatus:
push cx ;remember sign of result; 1 if result
; negative, 0 if result nonnegative
push ax ;remember high word of M1
mul bx ;high word M1 times high word M2
mov cx,ax ;accumulate result in CX:BX (BX not used
; until next operation, however)
;assume no overflow into DX
mov ax,si ;low word M1 times high word M2
mul bx
mov bx,ax
add cx,dx ;accumulate result in CX:BX
pop ax ;retrieve high word of M1
mul di ;high word M1 times low word M2
add bx,ax
adc cx,dx ;accumulate result in CX:BX
mov ax,si ;low word M1 times low word M2
muldi
if MUL-ROUNDING-ON
add ax,8000h ;round by adding 2^(-17)
adcbx,dx
else ;!MUL-ROUNDING-ON
add bx,dx, ;don't round
endif ;MUL-ROUNDING-ON
adc cx,0 ;accumulate result in CX:BX
mov dx,cx
mov ax,bx
pop cx
and cx,cx ;is the result negative?
jz FixedMulDone ;no, we're all set
neg dx ;yes, so negate DX:AX
neg ax
sbb dx,0
FixedMulDone:
pop di ;restore C register variables
pop si
endif;USE386
pop bp
ret
_FixedMul endp
;=====================================================================
; Divides one fixed-point value by another.
; C near-callable as:
; Fixedpoint FixedDiv(Fixedpoint Dividend, Fixedpoint Divisor);
FDparms struc
dw 2 dup(?) ;return address &amp pushed BP
Dividend dd?
Divisor dd?
FDparms ends
alignALIGNMENT
public_FixedDiv
_FixedDivproc near
pushbp
movbp,sp
if USE386
if DIV-ROUNDING-ON
sub cx,cx ;assume positive result
mov eax,[bp+Dividend]
and eax,eax ;positive dividend?
jns FDP1 ;yes
inc cx ;mark it's a negative dividend
neg eax ;make the dividend positive
FDP1: sub edx,edx ;make it a 64-bit dividend, then shift
; left 16 bits so that result will be in EAX
rol eax,16 ;put fractional part of dividend in
; high word of EAX
mov dx,ax ;put whole part of dividend in DX
sub ax,ax ;clear low word of EAX
mov ebx,dword ptr [bp+Divisor]
and ebx,ebx ;positive divisor?
jns FDP2 ;yes
dec cx ;mark it's a negative divisor
neg ebx ;make divisor positive
FDP2: div ebx ;divide
shr ebx,1 ;divisor/2, minus 1 if the divisor is
adc ebx,0 ; even
dec ebx
cmp ebx,edx ;set Carry if the remainder is at least
adc eax,0 ; half as large as the divisor, then
; use that to round up if necessary
and cx,cx ;should the result be made negative?
jz FDP3 ;no
neg eax ;yes, negate it
FDP3:
else ;!DIV-ROUNDING-ON
mov edx,[bp+Dividend]
sub eax,eax
shrd eax,edx,16 ;position so that result ends up
sar edx,16 ; in EAX
idiv dword ptr [bp+Divisor]
endif ;DIV-ROUNDING-ON
shld edx,eax,16 ;whole part of result in DX;
; fractional part is already in AX
else ;!USE386
;NOTE!!! Non-386 division uses a 32-bit dividend but only the upper 16 bits
; of the divisor; in other words, only the integer part of the divisor is
; used. This is done so that the division can be accomplished with two fast
; hardware divides instead of a slow software implementation, and is (in my
; opinion) acceptable because division is only used to project points to the
; screen (normally, the divisor is a Z coordinate), so there's no cumulative
; error, although there will be some error in pixel placement (the magnitude
; of the error is less the farther away from the Z=0 plane objects are). This
; is *not* a general-purpose divide, though; if the divisor is less than 1,
; for instance, a divide-by-zero error will result! For this reason, non-386
; projection can't be performed for points closer to the viewpoint than Z=1.
;figure out signs, so we can use
; unsigned divisions
subcx, cx ;assume both operands positive
mov ax,word ptr [bp+Dividend+2]
and ax,ax;first operand negative?
jns CheckSecondOperandD ;no
neg ax ;yes, so negate first operand
neg word ptr [bp+Dividend]
sbbax,0
inc cx ;mark that first operand is negative
CheckSecondOperandD:
mov bx,word ptr [bp+Divisor+2]
and bx,bx ;second operand negative?
jnsSaveSignStatusD;no
neg bx ;yes, so negate second operand
neg word ptr [bp+Divisor]
sbb bx,0
xor cx,1 ;mark that second operand is negative
SaveSignStatusD:
push cx ;remember sign of result; 1 if result
; negative, 0 if result nonnegative
sub dx,dx ;put Dividend+2 (integer part) in DX:AX
div bx ;first half of 32/16 division, integer part
; divided by integer part
mov cx,ax ;set aside integer part of result
mov ax,word ptr [bp+Dividend] ;concatenate the fractional part of
; the dividend to the remainder (fractional
; part) of the result from dividing the
; integer part of the dividend
div bx ;second half of 32/16 division
if DIV-ROUNDING-ON EQ 0
shr bx,1 ;divisor/2, minus 1 if the divisor is
adc bx,0 ; even
dec bx
cmp bx,dx ;set Carry if the remainder is at least
adc ax,0 ; half as large as the divisor, then
adc cx,0 ; use that to round up if necessary
endif ;DIV-ROUNDING-ON
mov dx,cx ;absolute value of result in DX:AX
pop cx
and cx,cx ;is the result negative?
jz FixedDivDone ;no, we're all set
neg dx ;yes, so negate DX:AX
neg ax
sbb dx,0
FixedDivDone:
endif ;USE386
pop bp
ret
_FixedDiv endp
;=====================================================================
; Returns the sine and cosine of an angle.
; C near-callable as:
; void CosSin(TAngle Angle, Fixedpoint *Cos, Fixedpoint *);
alignALIGNMENT
CosTable label dword
include costable.inc
SCparms struc
dw 2 dup(?) ;return address &amp pushed BP
Angle dw ? ;angle to calculate sine &amp cosine for
Cos dw ? ;pointer to cos destination
Sin dw ? ;pointer to sin destination
SCparms ends
alignALIGNMENT
public _CosSin
_CosSin procnear
push bp ;preserve stack frame
mov bp,sp ;set up local stack frame
if USE386
mov bx,[bp].Angle
and bx,bx ;make sure angle's between 0 and 2*pi
jns CheckInRange
MakePos: ;less than 0, so make it positive
add bx,360*10
js MakePos
jmp short CheckInRange
align ALIGNMENT
MakeInRange: ;make sure angle is no more than 2*pi
sub bx,360*10
CheckInRange:
cmp bx,360*10
jg MakeInRange
cmp bx,180*10 ;figure out which quadrant
ja BottomHalf ;quadrant 2 or 3
cmp bx,90*10 ;quadrant 0 or 1
ja Quadrant1
;quadrant 0
shl bx,2
move ax,CosTable[bx] ;look up sine
neg bx;sin(Angle) = cos(90-Angle)
move dx,CosTable[bx+90*10*4] ;look up cosine
jmp short CSDone
align ALIGNMENT
Quadrant1:
neg bx
add bx,180*10 ;convert to angle between 0 and 90
shl bx,2
move ax,CosTable[bx] ;look up cosine
neg eax ;negative in this quadrant
neg bx ;sin(Angle) = cos(90-Angle)
move dx,CosTable[bx+90*10*4] ;look up cosine
jmp short CSDone
align ALIGNMENT
BottomHalf: ;quadrant 2 or 3
neg bx
add bx,360*10 ;convert to angle between 0 and 180
cmp bx,90*10 ;quadrant 2 or 3
ja Quadrant2
;quadrant 3
shl bx,2
move ax,CosTable[bx] ;look up cosine
neg bx;sin(Angle) = cos(90-Angle)
move dx,CosTable[90*10*4+bx] ;look up sine
neg edx ;negative in this quadrant
jmp short CSDone
align ALIGNMENT
Quadrant2:
neg bx
add bx,180*10 ;convert to angle between 0 and 90
shl bx,2
move ax,CosTable[bx] ;look up cosine
neg eax ;negative in this quadrant
neg bx ;sin(Angle) = cos(90-Angle)
move dx,CosTable[90*10*4+bx] ;look up sine
neg edx ;negative in this quadrant
CSDone:
mov bx,[bp].Cos
mov [bx],eax
mov bx,[bp].Sin
mov [bx],edx
else ;!USE386
mov bx,[bp].Angle
and bx,bx ;make sure angle's between 0 and 2*pi
jns CheckInRange
MakePos: ;less than 0, so make it positive
add bx,360*10
js MakePos
jmp short CheckInRange
align ALIGNMENT
MakeInRange: ; make sure angle is no more than 2*pi
sub bx,360*10
CheckInRange:
cmp bx,360*10
jg MakeInRange
cmp bx,180*10 ;figure out which quadrant
ja BottomHalf ;quadrant 2 or 3
cmp bx,90*10 ;quadrant 0 or 1
jaQuadrant1
;quadrant 0
shl bx,2
mov ax,word ptr CosTable[bx] ;look up sine
mov dx,word ptr CosTable[bx+2]
neg bx ;sin(Angle) = cos(90-Angle)
mov cx,word ptr CosTable[bx+90*10*4+2] ;look up cosine
mov bx,word ptr CosTable[bx+90*10*4]
jmp CSDone
align ALIGNMENT
Quadrant1:
neg bx
add bx,180*10 ;convert to angle between 0 and 90
shl bx,2
mov ax,word ptr CosTable[bx] ;look up cosine
mov dx,word ptr CosTable[bx+2]
neg dx ;negative in this quadrant
neg ax
sbb dx,0
neg bx ;sin(Angle) = cos(90-Angle)
mov cx,word ptr CosTable[bx+90*10*4+2] ;look up cosine
mov bx,word ptr CosTable[bx+90*10*4]
jmp short CSDone
align ALIGNMENT
BottomHalf: ;quadrant 2 or 3
neg bx
add bx,360*10 ;convert to angle between 0 and 180
cmp bx,90*10 ;quadrant 2 or 3
jaQuadrant2
;quadrant 3
shl bx,2
mov ax,word ptr CosTable[bx] ;look up cosine
mov dx,word ptr CosTable[bx+2]
neg bx ;sin(Angle) = cos(90-Angle)
mov cx,word ptr CosTable[90*10*4+bx+2] ;look up sine
mov bx,word ptr CosTable[90*10*4+bx]
neg cx ;negative in this quadrant
neg bx
sbb cx,0
jmp short CSDone
align ALIGNMENT
Quadrant2:
neg bx
add bx,180*10 ;convert to angle between 0 and 90
shl bx,2
mov ax,word ptr CosTable[bx] ;look up cosine
mov dx,word ptr CosTable[bx+2]
neg dx ;negative in this quadrant
neg ax
sbb dx,0
neg bx ;sin(Angle) = cos(90-Angle)
mov cx,word ptr CosTable[90*10*4+bx+2] ;look up sine
mov bx,word ptr CosTable[90*10*4+bx]
neg cx ;negative in this quadrant
neg bx
sbb cx,0
CSDone:
push bx
mov bx,[bp].Cos
mov [bx],ax
mov [bx+2],dx
mov bx,[bp].Sin
pop ax
mov [bx],ax
mov [bx+2],cx
endif ;USE386
pop bp ;restore stack frame
ret
_CosSin endp
;=====================================================================
; Matrix multiplies Xform by SourceVec, and stores the result in
; DestVec. Multiplies a 4x4 matrix times a 4x1 matrix; the result
; is a 4x1 matrix. Cheats by assuming the W coord is 1 and the
; bottom row of the matrix is 0 0 0 1, and doesn't bother to set
; the W coordinate of the destination.
; C near-callable as:
; void XformVec(Xform WorkingXform, Fixedpoint *SourceVec,
; Fixedpoint *DestVec);
;
; This assembly code is equivalent to this C code:
; int i;
;
; for (i=0; i&lt;3; i++)
; DestVec[i] = FixedMul(WorkingXform[i][0], SourceVec[0]) +
; FixedMul(WorkingXform[i][1], SourceVec[1]) +
; FixedMul(WorkingXform[i][2], SourceVec[2]) +
; WorkingXform[i][3]; /* no need to multiply by W = 1 */
XVparms struc
dw 2 dup(?) ;return address &amp pushed BP
WorkingXform dw ? ;pointer to transform matrix
SourceVec dw ? ;pointer to source vector
DestVec dw ? ;pointer to destination vector
XVparms ends
; Macro for non-386 multiply. AX, BX, CX, DX destroyed.
FIXED-MUL MACRO M1,M2
local CheckSecondOperand,SaveSignStatus,FixedMulDone
;do four partial products and
; add them together, accumulating
; the result in CX:BX
;figure out signs, so we can use
; unsigned multiplies
sub cx,cx ;assume both operands positive
mov bx,word ptr [&ampM1&amp+2]
and bx,bx ;first operand negative?
jns CheckSecondOperand ;no
neg bx ;yes, so negate first operand
neg word ptr [&ampM1&amp]
sbb bx,0
mov word ptr [&ampM1&amp+2],bx
inc cx ;mark that first operand is negative
CheckSecondOperand:
mov bx,word ptr [&ampM2&amp+2]
and bx,bx ;second operand negative?
jns SaveSignStatus ;no
neg bx ;yes, so negate second operand
neg word ptr [&ampM2&amp]
sbb bx,0
mov word ptr [&ampM2&amp+2],bx
xor cx,1 ;mark that second operand is negative
SaveSignStatus:
push cx ;remember sign of result; 1 if result
; negative, 0 if result nonnegative
mov ax,word ptr [&ampM1&amp+2] ;high word times high word
mul word ptr [&ampM2&amp+2]
mov cx,ax ;
;assume no overflow into DX
mov ax,word ptr [&ampM1&amp+2] ;high word times low word
mul word ptr [&ampM2&amp]
mov bx,ax
add cx,dx
mov ax,word ptr [&ampM1&amp] ;low word times high word
mul word ptr [&ampM2&amp+2]
add bx,ax
adc cx,dx
mov ax,word ptr [&ampM1&amp] ;low word times low word
mul word ptr [&ampM2&amp]
if MUL-ROUNDING-ON
add ax,8000h ;round by adding 2^(-17)
adc bx,dx
else ;!MUL-ROUNDING-ON
add bx,dx ;don't round
endif ;MUL-ROUNDING-ON
adc cx,0
mov dx,cx
mov ax,bx
pop cx
and cx,cx ;is the result negative?
jz FixedMulDone ;no, we're all set
neg dx ;yes, so negate DX:AX
neg ax
sbb dx,0
FixedMulDone:
ENDM
align ALIGNMENT
public _XformVec
_XformVecprocnear
push bp ;preserve stack frame
mov bp,sp ;set up local stack frame
push si ;preserve register variables
push di
if USE386
mov si,[bp].WorkingXform ;SI points to xform matrix
mov bx,[bp].SourceVec ;BX points to source vector
mov di,[bp].DestVec ;DI points to dest vector
soff=0
doff=0
REPT 3 ;do once each for dest X, Y, and Z
mov eax,[si+soff] ;column 0 entry on this row
imul dword ptr [bx] ;xform entry times source X entry
if MUL-ROUNDING-ON
add eax,8000h ;round by adding 2^(-17)
adc edx,0 ;whole part of result is in DX
endif ;MUL-ROUNDING-ON
shrd eax,edx,16 ;shift the result back to 16.16 form
move cx,eax ;set running total
move ax,[si+soff+4] ;column 1 entry on this row
imul dword ptr [bx+4] ;xform entry times source Y entry
if MUL-ROUNDING-ON
add eax,8000h ;round by adding 2^(-17)
adc edx,0 ;whole part of result is in DX
endif ;MUL-ROUNDING-ON
shrd eax,edx,16 ;shift the result back to 16.16 form
add ecx,eax ;running total for this row
move ax,[si+soff+8] ;column 2 entry on this row
imul dword ptr [bx+8] ;xform entry times source Z entry
if MUL-ROUNDING-ON
add eax,8000h ;round by adding 2^(-17)
adc edx,0 ;whole part of result is in DX
endif ;MUL-ROUNDING-ON
shrd eax,edx,16 ;shift the result back to 16.16 form
add ecx,eax ;running total for this row
add ecx,[si+soff+12] ;add in translation
mov [di+doff],ecx ;save the result in the dest vector
soff=soff+16
doff=doff+4
ENDM
else ;!USE386
mov si,[bp].WorkingXform ;SI points to xform matrix
mov di,[bp].SourceVec ;DI points to source vector
mov bx,[bp].DestVec ;BX points to dest vector
push bp ;preserve stack frame pointer
soff=0
doff=0
REPT 3 ;do once each for dest X, Y, and Z
push bx ;remember dest vector pointer
push word ptr [si+soff+2]
push word ptr [si+soff]
push word ptr [di+2]
push word ptr [di]
call _FixedMul ;xform entry times source X entry
add sp,8;clear parameters from stack
mov cx,ax ;set running total
mov bp,dx
push cx ;preserve low word of running total
push word ptr [si+soff+4+2]
push word ptr [si+soff+4]
push word ptr [di+4+2]
push word ptr [di+4]
call _FixedMul ;xform entry times source Y entry
add sp,8 ;clear parameters from stack
pop cx ;restore low word of running total
add cx,ax ;running total for this row
adc bp,dx
push cx ;preserve low word of running total
push word ptr [si+soff+8+2]
push word ptr [si+soff+8]
push word ptr [di+8+2]
push word ptr [di+8]
call _FixedMul ;xform entry times source Z entry
add sp,8 ;clear parameters from stack
pop cx ;restore low word of running total
add cx,ax ;running total for this row
adc bp,dx
add cx,[si+soff+12] ;add in translation
adc bp,[si+soff+12+2]
pop bx ;restore dest vector pointer
mov [bx+doff],cx ;save the result in the dest vector
mov [bx+doff+2],bp
soff=soff+16
doff=doff+4
ENDM
pop bp ;restore stack frame pointer
endif ;USE386
pop di ;restore register variables
pop si
pop bp ;restore stack frame
ret
_XformVecendp
;=====================================================================
; Matrix multiplies SourceXform1 by SourceXform2 and stores the
; result in DestXform. Multiplies a 4x4 matrix times a 4x4 matrix;
; the result is a 4x4 matrix. Cheats by assuming the bottom row of
; each matrix is 0 0 0 1, and doesn't bother to set the bottom row
; of the destination.
; C near-callable as:
; void ConcatXforms(Xform SourceXform1, Xform SourceXform2,
; Xform DestXform)
;
; This assembly code is equivalent to this C code:
; int i, j;
;
; for (i=0; i&lt;3; i++) {
; for (j=0; j&lt;3; j++)
; DestXform[i][j] =
; FixedMul(SourceXform1[i][0], SourceXform2[0][j]) +
; FixedMul(SourceXform1[i][1], SourceXform2[1][j]) +
; FixedMul(SourceXform1[i][2], SourceXform2[2][j]);
; DestXform[i][3] =
; FixedMul(SourceXform1[i][0], SourceXform2[0][3]) +
; FixedMul(SourceXform1[i][1], SourceXform2[1][3]) +
; FixedMul(SourceXform1[i][2], SourceXform2[2][3]) +
; SourceXform1[i][3];
; }
CXparms struc
dw 2 dup(?) ;return address &amp pushed BP
SourceXform1 dw ? ;pointer to first source xform matrix
SourceXform2 dw ? ;pointer to second source xform matrix
DestXform dw ? ;pointer to destination xform matrix
CXparms ends
align ALIGNMENT
public _ConcatXforms
_ConcatXformsprocnear
push bp ;preserve stack frame
mov bp,sp ;set up local stack frame
push si ;preserve register variables
push di
if USE386
mov bx,[bp].SourceXform2 ;BX points to xform2 matrix
mov si,[bp].SourceXform1 ;SI points to xform1 matrix
mov di,[bp].DestXform ;DI points to dest xform matrix
roff=0 ;row offset
REPT 3 ;once for each row
coff=0 ;column offset
REPT 3 ;once for each of the first 3 columns,
; assuming 0 as the bottom entry (no
; translation)
move ax,[si+roff] ;column 0 entry on this row
imul dword ptr [bx+coff] ;times row 0 entry in column
if MUL-ROUNDING-ON
add eax,8000h ;round by adding 2^(-17)
adc edx,0 ;whole part of result is in DX
endif ;MUL-ROUNDING-ON
shrd eax,edx,16 ;shift the result back to 16.16 form
move cx,eax ;set running total
move ax,[si+roff+4] ;column 1 entry on this row
imuld word ptr [bx+coff+16] ;times row 1 entry in col
if MUL-ROUNDING-ON
add eax,8000h ;round by adding 2^(-17)
adc edx,0 ;whole part of result is in DX
endif ;MUL-ROUNDING-ON
shrd eax,edx,16 ;shift the result back to 16.16 form
add ecx,eax ;running total
move ax,[si+roff+8] ;column 2 entry on this row
imuld word ptr [bx+coff+32] ;times row 2 entry in col
if MUL-ROUNDING-ON
add eax,8000h ;round by adding 2^(-17)
adc edx,0 ;whole part of result is in DX
endif ;MUL-ROUNDING-ON
shrd eax,edx,16 ;shift the result back to 16.16 form
add ecx,eax ;running total
mov [di+coff+roff],ecx ;save the result in dest matrix
coff=coff+4 ;point to next col in xform2 &amp dest
ENDM
;now do the fourth column, assuming
; 1 as the bottom entry, causing
; translation to be performed
move ax,[si+roff] ;column 0 entry on this row
imuld word ptr [bx+coff] ;times row 0 entry in column
if MUL-ROUNDING-ON
add eax,8000h ;round by adding 2^(-17)
adc edx,0 ;whole part of result is in DX
endif ;MUL-ROUNDING-ON
shrd eax,edx,16 ;shift the result back to 16.16 form
move cx,eax ;set running total
move ax,[si+roff+4] ;column 1 entry on this row
imuld word ptr [bx+coff+16] ;times row 1 entry in col
if MUL-ROUNDING-ON
add eax,8000h ;round by adding 2^(-17)
adc edx,0 ;whole part of result is in DX
endif ;MUL-ROUNDING-ON
shrd eax,edx,16 ;shift the result back to 16.16 form
add ecx,eax ;running total
move ax,[si+roff+8] ;column 2 entry on this row
imuld word ptr [bx+coff+32] ;times row 2 entry in col
if MUL-ROUNDING-ON
add eax,8000h ;round by adding 2^(-17)
adc edx,0 ;whole part of result is in DX
endif ;MUL-ROUNDING-ON
shrd eax,edx,16 ;shift the result back to 16.16 form
add ecx,eax ;running total
add ecx,[si+roff+12] ;add in translation
mov [di+coff+roff],ecx ;save the result in dest matrix
coff=coff+4 ;point to next col in xform2 &amp dest
roff=roff+16 ;point to next col in xform2 &amp dest
ENDM
else ;!USE386
mov di,[bp].SourceXform2 ;DI points to xform2 matrix
mov si,[bp].SourceXform1 ;SI points to xform1 matrix
mov bx,[bp].DestXform ;BX points to dest xform matrix
push bp ;preserve stack frame pointer
roff=0 ;row offset
REPT 3 ;once for each row
coff=0 ;column offset
REPT 3 ;once for each of the first 3 columns,
; assuming 0 as the bottom entry (no
; translation)
push bx ;remember dest vector pointer
push word ptr [si+roff+2]
push word ptr [si+roff]
push word ptr [di+coff+2]
push word ptr [di+coff]
call _FixedMul ;column 0 entry on this row times row 0
; entry in column
addsp,8;clear parameters from stack
mov cx,ax ;set running total
mov bp,dx
push cx ;preserve low word of running total
push word ptr [si+roff+4+2]
push word ptr [si+roff+4]
push word ptr [di+coff+16+2]
push word ptr [di+coff+16]
call _FixedMul ;column 1 entry on this row times row 1
; entry in column
add sp,8 ;clear parameters from stack
pop cx ;restore low word of running total
add cx,ax ;running total for this row
adc bp,dx
push cx ;preserve low word of running total
push word ptr [si+roff+8+2]
push word ptr [si+roff+8]
push word ptr [di+coff+32+2]
push word ptr [di+coff+32]
call _FixedMul ;column 1 entry on this row times row 1
; entry in column
add sp,8 ;clear parameters from stack
pop cx ;restore low word of running total
add cx,ax ;running total for this row
adc bp,dx
pop bx ;restore DestXForm pointer
mov [bx+coff+roff],cx ;save the result in dest matrix
mov [bx+coff+roff+2],bp
coff=coff+4 ;point to next col in xform2 &amp dest
ENDM
;now do the fourth column, assuming
; 1 as the bottom entry, causing
; translation to be performed
push bx ;remember dest vector pointer
push word ptr [si+roff+2]
push word ptr [si+roff]
push word ptr [di+coff+2]
push word ptr [di+coff]
call _FixedMul ;column 0 entry on this row times row 0
; entry in column
add sp,8 ;clear parameters from stack
mov cx,ax ;set running total
mov bp,dx
push cx ;preserve low word of running total
push word ptr [si+roff+4+2]
push word ptr [si+roff+4]
push word ptr [di+coff+16+2]
push word ptr [di+coff+16]
call _FixedMul ;column 1 entry on this row times row 1
; entry in column
add sp,8 ;clear parameters from stack
pop cx ;restore low word of running total
add cx,ax ;running total for this row
adc bp,dx
push cx ;preserve low word of running total
push word ptr [si+roff+8+2]
push word ptr [si+roff+8]
push word ptr [di+coff+32+2]
push word ptr [di+coff+32]
call _FixedMul ;column 1 entry on this row times row 1
; entry in column
add sp,8 ;clear parameters from stack
pop cx ;restore low word of running total
add cx,ax ;running total for this row
adc bp,dx
add cx,[si+roff+12] ;add in translation
add bp,[si+roff+12+2]
pop bx ;restore DestXForm pointer
mov [bx+coff+roff],cx ;save the result in dest matrix
mov [bx+coff+roff+2],bp
coff=coff+4 ;point to next col in xform2 &amp dest
roff=roff+16 ;point to next col in xform2 &amp dest
ENDM
pop bp ;restore stack frame pointer
endif ;USE386
pop di ;restore register variables
pop si
pop bp ;restore stack frame
ret
_ConcatXforms endp
end
</PRE>
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<TITLE>Michael Abrash's Graphics Programming Black Book Special Edition: 3-D Shading</TITLE>
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<H4 ALIGN="LEFT"><A NAME="Heading4"></A><FONT COLOR="#000077">Shading</FONT></H4>
<P>So far, the polygons out of which our animated objects have been built have had colors of fixed intensities. For example, a face of a cube might be blue, or green, or white, but whatever color it is, that color never brightens or dims. Fixed colors are easy to implement, but they don&#146;t make for very realistic animation. In the real world, the intensity of the color of a surface varies depending on how brightly it is illuminated. The ability to simulate the illumination of a surface, or shading, is the next feature we&#146;ll add to X-Sharp.
</P>
<P>The overall shading of an object is the sum of several types of shading components. <I>Ambient shading</I> is illumination by what you might think of as background light, light that&#146;s coming from all directions; all surfaces are equally illuminated by ambient light, regardless of their orientation. <I>Directed lighting</I>, producing diffuse shading, is illumination from one or more specific light sources. Directed light has a specific direction, and the angle at which it strikes a surface determines how brightly it lights that surface. <I>Specular reflection</I> is the tendency of a surface to reflect light in a mirrorlike fashion. There are other sorts of shading components, including transparency and atmospheric effects, but the ambient and diffuse-shading components are all we&#146;re going to deal with in X-Sharp.</P>
<H4 ALIGN="CENTER"><A NAME="Heading5"></A><FONT COLOR="#000077">Ambient Shading</FONT></H4>
<P>The basic model for both ambient and diffuse shading is a simple one. Each surface has a reflectivity between 0 and 1, where 0 means all light is absorbed and 1 means all light is reflected. A certain amount of light energy strikes each surface. The energy (intensity) of the light is expressed such that if light of intensity 1 strikes a surface with reflectivity 1, then the brightest possible shading is displayed for that surface. Complicating this somewhat is the need to support color; we do this by separating reflectance and shading into three components each&#151;red, green, and blue&#151;and calculating the shading for each color component separately for each surface.
</P>
<P>Given an ambient-light red intensity of IA<SUB><SMALL>red</SMALL></SUB> and a surface red reflectance R<SUB><SMALL>red</SMALL></SUB>, the displayed red ambient shading for that surface, as a fraction of the maximum red intensity, is simply min(IA<SUB><SMALL>red</SMALL></SUB>&#215; R<SUB><SMALL>red</SMALL></SUB>, 1). The green and blue color components are handled similarly. That&#146;s really all there is to ambient shading, although of course we must design some way to map displayed color components into the available palette of colors; I&#146;ll do that in the next chapter. Ambient shading isn&#146;t the whole shading picture, though. In fact, scenes tend to look pretty bland without diffuse shading.</P>
<H4 ALIGN="CENTER"><A NAME="Heading6"></A><FONT COLOR="#000077">Diffuse Shading</FONT></H4>
<P>Diffuse shading is more complicated than ambient shading, because the effective intensity of directed light falling on a surface depends on the angle at which it strikes the surface. According to Lambert&#146;s law, the light energy from a directed light source striking a surface is proportional to the cosine of the angle at which it strikes the surface, with the angle measured relative to a vector perpendicular to the polygon (a polygon normal), as shown in Figure 54.1. If the red intensity of directed light is ID<SUB><SMALL>red</SMALL></SUB>, the red reflectance of the surface is R<SUB><SMALL>red</SMALL></SUB>, and the angle between the incoming directed light and the surface&#146;s normal is theta, then the displayed red diffuse shading for that surface, as a fraction of the largest possible red intensity, is min (ID<SUB><SMALL>red</SMALL></SUB>&#215;R<SUB><SMALL>red</SMALL></SUB>&#215;cos(&#952;), 1).</P>
<P>That&#146;s easy enough to calculate&#151;but seemingly slow. Determining the cosine of an angle can be sped up with a table lookup, but there&#146;s also the task of figuring out the angle, and, all in all, it doesn&#146;t seem that diffuse shading is going to be speedy enough for our purposes. Consider this, however: According to the properties of the dot product (denoted by the operator &#147;&#149;&#148;, as shown in Figure 54.2), cos(q)=(v&#149;w)/ |v| &#215; |w| ), where v and w are vectors, q is the angle between v and w, and |v| is the length of v. Suppose, now, that v and w are unit vectors; that is, vectors exactly one unit long. Then the above equation reduces to cos(q)=v&#149;w. In other words, we can calculate the cosine between N, the unit-normal vector (one-unit-long perpendicular vector) of a polygon, and L', the reverse of a unit vector describing the direction of a light source, with just three multiplies and two adds. (I&#146;ll explain why the light-direction vector must be reversed later.) Once we have that, we can easily calculate the red diffuse shading from a directed light source as min(ID<SUB><SMALL>red</SMALL></SUB>&#215;R<SUB><SMALL>red</SMALL></SUB>&#215;(L'&#149; N), 1) and likewise for the green and blue color components.</P>
<P><A NAME="Fig1"><!-- </A><A HREF="javascript:displayWindow('images/54-01.jpg',406,116 )"> --><IMG SRC="images/54-01.jpg"><BR><!-- </A>
<BR><A HREF="javascript:displayWindow('images/54-01.jpg',406,116)"> --><FONT COLOR="#000077"><B>Figure 54.1</B></FONT></A>&nbsp;&nbsp;<I>Illumination by a directed light source</I>
</P>
<P><A NAME="Fig2"><!-- </A><A HREF="javascript:displayWindow('images/54-02.jpg',409,86 )"> --><IMG SRC="images/54-02.jpg"><BR><!-- </A>
<BR><A HREF="javascript:displayWindow('images/54-02.jpg',409,86)"> --><FONT COLOR="#000077"><B>Figure 54.2</B></FONT></A>&nbsp;&nbsp;<I>The dot product of two vectors.</I>
</P>
<P>The overall red shading for each polygon can be calculated by summing the ambient-shading red component with the diffuse-shading component from each light source, as in min((IA<SUB><SMALL>red</SMALL></SUB>&#215;R<SUB><SMALL>red</SMALL></SUB>) + (ID<SUB><SMALL>red0</SMALL></SUB>&#215;R<SUB><SMALL>red</SMALL></SUB>&#215;(L<SUB><SMALL>0</SMALL></SUB>' &#149; N)) + (ID<SUB><SMALL>red1</SMALL></SUB>&#215;R<SUB><SMALL>red</SMALL></SUB>&#215;(L<SUB><SMALL>1</SMALL></SUB>' &#149; N)) +..., 1) where ID<SUB><SMALL>red0</SMALL></SUB> and L<SUB><SMALL>0</SMALL></SUB>' are the red intensity and the reversed unit-direction vector, respectively, for spotlight 0. Listing 54.2 shows the X-Sharp module DRAWPOBJ.C, which performs ambient and diffuse shading. Toward the end, you will find the code that performs shading exactly as described by the above equation, first calculating the ambient red, green, and blue shadings, then summing that with the diffuse red, green, and blue shadings generated by each directed light source.</P><P><BR></P>
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<P><B>LISTING 54.2 DRAWPOBJ.C</B></P>
<!-- CODE //-->
<PRE>
/* Draws all visible faces in the specified polygon-based object. The object
must have previously been transformed and projected, so that all vertex
arrays are filled in. Ambient and diffuse shading are supported. */
#include "polygon.h"
void DrawPObject(PObject * ObjectToXform)
{
int i, j, NumFaces = ObjectToXform-&gt;NumFaces, NumVertices;
int * VertNumsPtr, Spot;
Face * FacePtr = ObjectToXform-&gt;FaceList;
Point * ScreenPoints = ObjectToXform-&gt;ScreenVertexList;
PointListHeader Polygon;
Fixedpoint Diffusion;
ModelColor ColorTemp;
ModelIntensity IntensityTemp;
Point3 UnitNormal, *NormalStartpoint, *NormalEndpoint;
long v1, v2, w1, w2;
Point Vertices[MAX-POLY-LENGTH];
/* Draw each visible face (polygon) of the object in turn */
for (i=0; i&lt;NumFaces; i++, FacePtr++) {
/* Remember where we can find the start and end of the polygon's
unit normal in view space, and skip over the unit normal endpoint
entry. The end and start points of the unit normal to the polygon
must be the first and second entries in the polgyon's vertex list.
Note that the second point is also an active polygon vertex */
VertNumsPtr = FacePtr-&gt;VertNums;
NormalEndpoint = &ampObjectToXform-&gt;XformedVertexList[*VertNumsPtr++];
NormalStartpoint = &ampObjectToXform-&gt;XformedVertexList[*VertNumsPtr];
/* Copy over the face's vertices from the vertex list */
NumVertices = FacePtr-&gt;NumVerts;
for (j=0; j&lt;NumVertices; j++)
Vertices[j] = ScreenPoints[*VertNumsPtr++];
/* Draw only if outside face showing (if the normal to the polygon
in screen coordinates points toward the viewer; that is, has a
positive Z component) */
v1 = Vertices[1].X - Vertices[0].X;
w1 = Vertices[NumVertices-1].X - Vertices[0].X;
v2 = Vertices[1].Y - Vertices[0].Y;
w2 = Vertices[NumVertices-1].Y - Vertices[0].Y;
if ((v1*w2 - v2*w1) &gt; 0) {
/* It is facing the screen, so draw */
/* Appropriately adjust the extent of the rectangle used to
erase this object later */
for (j=0; j&lt;NumVertices; j++) {
if (Vertices[j].X &gt;
ObjectToXform-&gt;EraseRect[NonDisplayedPage].Right)
if (Vertices[j].X &lt; SCREEN-WIDTH)
ObjectToXform-&gt;EraseRect[NonDisplayedPage].Right =
Vertices[j].X;
else ObjectToXform-&gt;EraseRect[NonDisplayedPage].Right =
SCREEN-WIDTH;
if (Vertices[j].Y &gt;
ObjectToXform-&gt;EraseRect[NonDisplayedPage].Bottom)
if (Vertices[j].Y &lt; SCREEN-HEIGHT)
ObjectToXform-&gt;EraseRect[NonDisplayedPage].Bottom =
Vertices[j].Y;
else ObjectToXform-&gt;EraseRect[NonDisplayedPage].Bottom=
SCREEN-HEIGHT;
if (Vertices[j].X &lt;
ObjectToXform-&gt;EraseRect[NonDisplayedPage].Left)
if (Vertices[j].X &gt; 0)
ObjectToXform-&gt;EraseRect[NonDisplayedPage].Left =
Vertices[j].X;
else ObjectToXform-&gt;EraseRect[NonDisplayedPage].Left=0;
if (Vertices[j].Y &lt;
ObjectToXform-&gt;EraseRect[NonDisplayedPage].Top)
if (Vertices[j].Y &gt; 0)
ObjectToXform-&gt;EraseRect[NonDisplayedPage].Top =
Vertices[j].Y;
else ObjectToXform-&gt;EraseRect[NonDisplayedPage].Top=0;
}
/* See if there's any shading */
if (FacePtr-&gt;ShadingType == 0) {
/* No shading in effect, so just draw */
DRAW-POLYGON(Vertices, NumVertices, FacePtr-&gt;ColorIndex, 0, 0);
} else {
/* Handle shading */
/* Do ambient shading, if enabled */
if (AmbientOn &amp&amp (FacePtr-&gt;ShadingType &amp AMBIENT-SHADING)) {
/* Use the ambient shading component */
IntensityTemp = AmbientIntensity;
} else {
SET-INTENSITY(IntensityTemp, 0, 0, 0);
}
/* Do diffuse shading, if enabled */
if (FacePtr-&gt;ShadingType &amp DIFFUSE-SHADING) {
/* Calculate the unit normal for this polygon, for use in dot
products */
UnitNormal.X = NormalEndpoint-&gt;X - NormalStartpoint-&gt;X;
UnitNormal.Y = NormalEndpoint-&gt;Y - NormalStartpoint-&gt;Y;
UnitNormal.Z = NormalEndpoint-&gt;Z - NormalStartpoint-&gt;Z;
/* Calculate the diffuse shading component for each active
spotlight */
for (Spot=0; Spot&lt;MAX-SPOTS; Spot++) {
if (SpotOn[Spot] != 0) {
/* Spot is on, so sum, for each color component, the
intensity, accounting for the angle of the light rays
relative to the orientation of the polygon */
/* Calculate cosine of angle between the light and the
polygon normal; skip if spot is shining from behind
the polygon */
if ((Diffusion = DOT-PRODUCT(SpotDirectionView[Spot],
UnitNormal)) &gt; 0) {
IntensityTemp.Red +=
FixedMul(SpotIntensity[Spot].Red, Diffusion);
IntensityTemp.Green +=
FixedMul(SpotIntensity[Spot].Green, Diffusion);
IntensityTemp.Blue +=
FixedMul(SpotIntensity[Spot].Blue, Diffusion);
}
}
}
}
/* Convert the drawing color to the desired fraction of the
brightest possible color */
IntensityAdjustColor(&ampColorTemp, &ampFacePtr-&gt;FullColor,
&ampIntensityTemp);
/* Draw with the cumulative shading, converting from the general
color representation to the best-match color index */
DRAW-POLYGON(Vertices, NumVertices,
ModelColorToColorIndex(&ampColorTemp), 0, 0);
}
}
}
}
</PRE>
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<TITLE>Michael Abrash's Graphics Programming Black Book Special Edition: 3-D Shading</TITLE>
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<H4 ALIGN="LEFT"><A NAME="Heading7"></A><FONT COLOR="#000077">Shading: Implementation Details</FONT></H4>
<P>In order to calculate the cosine of the angle between an incoming light source and a polygon&#146;s unit normal, we must first have the polygon&#146;s unit normal. This could be calculated by generating a cross-product on two polygon edges to generate a normal, then calculating the normal&#146;s length and scaling to produce a unit normal. Unfortunately, that would require taking a square root, so it&#146;s not a desirable course of action. Instead, I&#146;ve made a change to X-Sharp&#146;s polygon format. Now, the first vertex in a shaded polygon&#146;s vertex list is the end-point of a unit normal that starts at the second point in the polygon&#146;s vertex list, as shown in Figure 54.3. The first point isn&#146;t one of the polygon&#146;s vertices, but is used only to generate a unit normal. The second point, however, is a polygon vertex. Calculating the difference vector between the first and second points yields the polygon&#146;s unit normal. Adding a unit-normal endpoint to each polygon isn&#146;t free; each of those end-points has to be transformed, along with the rest of the vertices, and that takes time. Still, it&#146;s faster than calculating a unit normal for each polygon from scratch.
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<P><A NAME="Fig3"><!-- </A><A HREF="javascript:displayWindow('images/54-03.jpg',409,134 )"> --><IMG SRC="images/54-03.jpg"><BR><!-- </A>
<BR><A HREF="javascript:displayWindow('images/54-03.jpg',409,134)"> --><FONT COLOR="#000077"><B>Figure 54.3</B></FONT></A>&nbsp;&nbsp;<I>The unit normal in the polygon data structure.</I>
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<P><A NAME="Fig4"><!-- </A><A HREF="javascript:displayWindow('images/54-04.jpg',408,142 )"> --><IMG SRC="images/54-04.jpg"><BR><!-- </A>
<BR><A HREF="javascript:displayWindow('images/54-04.jpg',408,142)"> --><FONT COLOR="#000077"><B>Figure 54.4</B></FONT></A>&nbsp;&nbsp;<I>The reversed light source vector.</I>
</P>
<P>We also need a unit vector for each directed light source. The directed light sources I&#146;ve implemented in X-Sharp are spotlights; that is, they&#146;re considered to be point light sources that are infinitely far away. This allows the simplifying assumption that all light rays from a spotlight are parallel and of equal intensity throughout the displayed universe, so each spotlight can be represented with a single unit vector and a single intensity. The only trick is that in order to calculate the desired cos(theta) between the polygon unit normal and a spotlight&#146;s unit vector, the direction of the spotlight&#146;s unit vector must be reversed, as shown in Figure 54.4. This is necessary because the dot product implicitly places vectors with their start points at the same location when it&#146;s used to calculate the cosine of the angle between two vectors. The light vector is incoming to the polygon surface, and the unit normal is outbound, so only by reversing one vector or the other will we get the cosine of the desired angle.
</P>
<P>Given the two unit vectors, it&#146;s a piece of cake to calculate intensities, as shown in Listing 54.2. The sample program DEMO1, in the X-Sharp archive on the listings disk (built by running K1.BAT), puts the shading code to work displaying a rotating ball with ambient lighting and three spot lighting sources that the user can turn on and off. What you&#146;ll see when you run DEMO1 is that the shading is very good&#151;face colors change very smoothly indeed&#151;so long as only green lighting sources are on. However, if you combine spotlight two, which is blue, with any other light source, polygon colors will start to shift abruptly and unevenly. As configured in the demo, the palette supports a wide range of shading intensities for a pure version of any one of the three primary colors, but a very limited number of intensity steps (four, in this case) for each color component when two or more primary colors are mixed. While this situation can be improved, it is fundamentally a result of the restricted capabilities of the 256-color palette, and there is only so much that can be done without a larger color set. In the next chapter, I&#146;ll talk about some ways to improve the quality of 256-color shading.</P><P><BR></P>
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